Titles i n Thi s Serie s

43 Lui s A . Caffarell i an d Xavie r Cabr6 , Full y nonlinea r ellipti c equations , 199 5

42 Victo r Guil lemi n an d Shlom o Sternberg , Variation s o n a them e b y Kepler , 199 0

41 Alfre d Tarsk i an d Steve n Givant , A formalizatio n o f se t theor y withou t variables ,

1987

40 R . H . Bing , Th e geometri c topolog y o f 3-manifolds , 198 3

39 N . Jacobson , Structur e an d representation s o f Jorda n algebras , 196 8

38 O . Ore , Theor y o f graphs , 196 2

37 N . Jacobson , Structur e o f rings , 195 6

36 W . H . Gottschal k an d G . A . Hedlund , Topologica l dynamics , 195 5

35 A . C . SchaefFe r an d D . C . Spencer , Coefficien t region s fo r Schlich t functions , 195 0

34 J . L . Walsh , Th e locatio n o f critica l point s o f analyti c an d harmoni c functions , 195 0

33 J . F . R i t t , Differentia l algebra , 195 0

32 R . L . Wi lder , Topolog y o f manifolds , 194 9

31 E . Hil l e an d R . S . Phil l ips , Functiona l analysi s an d semigroups , 195 7

30 T . Rad6 , Lengt h an d area , 194 8

29 A . Weil , Foundation s o f algebrai c geometry , 194 6

28 G . T . W h y b u r n , Analyti c topology , 194 2

27 S . Lefschetz , Algebrai c topology , 194 2

26 N . Levinson , Ga p an d densit y theorems , 194 0

25 Garret t Birkhoff , Lattic e theory , 194 0

24 A . A . Albert , Structur e o f algebras , 193 9

23 G . Szego , Orthogona l polynomials , 193 9

22 C . N . Moore , Summabl e serie s an d convergenc e factors , 193 8

21 J . M . T h o m a s , Differentia l systems , 193 7

20 J . L . Walsh , Interpolatio n an d approximatio n b y rationa l function s i n th e comple x

domain, 193 5

19 R . E . A . C . Pa le y an d N . Wiener , Fourie r transform s i n th e comple x domain , 193 4

18 M . Morse , Th e calculu s o f variation s i n th e large , 193 4

17 J . M . Wedderburn , Lecture s o n matrices , 193 4

16 G . A . Bl iss , Algebrai c functions , 193 3

15 M . H . Stone , Linea r transformation s i n Hilber t spac e an d thei r application s t o

analysis, 193 2

14 J . F . R i t t , Differentia l equation s fro m th e algebrai c standpoint , 193 2

13 R . L . Moore , Foundation s o f poin t se t theory , 193 2

12 S . Lefschetz , Topology , 193 0

11 D . Jackson , Th e theor y o f approximation , 193 0

10 A . B . Coble , Algebrai c geometr y an d thet a functions , 192 9

9 G . D . Birkhoff , Dynamica l systems , 192 7

8 L . P . E i s e n h a r t , Non-Riemannia n geometry , 192 2

7 E . T . Bell , Algebrai c arithmetic , 192 7 6 G . C . Evans , Th e logarithmi c potential , discontinuou s Dirichle t an d Neuman n

problems, 192 7 5.1 G . C . Evans , Functional s an d thei r applications ; selecte d topics , includin g integra l

equations, 191 8

5.2 O . Veblen , Analysi s situs , 192 2

4 L . E . Dickson , O n invariant s an d th e theor y o f number s

(Continued in the back of this publication)

This page intentionally left blank

AMERICAN MATHEMATICAL SOCIETY COLLOQUIUM PUBLICATIONS VOLUME 18

The Calculus of Variations in the Large Marston Morse

American Mathematical Society Providence, Rhode Island

http://dx.doi.org/10.1090/coll/018

2000 Mathematics Subject Classification. Primar y 49-02 ; Secondary 58E30 , 58E05 .

International Standard Seria l Number 0065-9258 International Standar d Boo k Number 0-8218-1018- 9 Library o f Congress Catalog Card Numbe r 34-4090 9

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t libraries actin g fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o copy a chapte r fo r us e i n teachin g o r research . Permissio n i s grante d t o quot e brie f passages from thi s publication in reviews , provide d the customary acknowledgmen t o f the source i s given.

Republication, systematic copying, or multiple reproduction of any material i n thi s publication i s permitted only unde r license fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can als o be made by e-mai l t o reprint-permissionfcams.org .

© Copyrigh t 193 4 by the American Mathematica l Society . Printed i n the Unite d State s of America .

The American Mathematica l Societ y retain s al l right s except thos e granted t o the Unite d State s Government .

@ Th e paper use d i n this book is acid-free an d fail s within the guideline s established t o ensure permanence an d durability .

Visit the AMS home page at URL : http://www.arns.org /

14 13 12 11 10 0 6 05 04 03 02 01

FOREWORD

For several years the research of the writer has been oriented by a conception of what might be termed macro-analysis. I t seems probable to the author tha t many of the objectively important problems in mathematical physics , geometry, and analysis cannot be solved without radical additions to the methods of what is now strictly regarde d a s pure analysis . An y problem whic h i s non-linear in character, whic h involves more than on e coordinat e syste m o r more tha n on e variable, or whose structure i s initially defined i n the large , is likely to requir e considerations o f topology an d grou p theory i n orde r t o arriv e a t it s meaning and it s solution . I n th e solutio n o f suc h problem s classica l analysi s wil l fre -quently appear as an instrument in the small, integrated over the whole problem with the aid of group theory or topology. Suc h conceptions are not due to the author. I t wil l be sufficient t o say that Henri Poincar6 was among the first to have a consciou s theor y o f macro-analysis , an d o f al l mathematician s wa s doubtless the one who most effectively pu t such a theory into practice.

The principal contribution of the author has been first to giv e a n analysi s in the large o f a functio n / o f m variables , an d the n t o exten d thi s analysi s t o functionals. Th e functionals chose n have been those of the Calculus of Varia -tions. Althoug h ther e ar e indication s tha t furthe r dee p extension s t o othe r functionals exist , suc h extension s ar e beyon d th e scop e o f thes e Lectures . Whereas the analogies between the theory of linear and quadratic forms and the theory of functionals hav e been well recognized sinc e the wor k of Hilbert , th e analogies in the large between functions and functionals here presented have not been s o recognized , an d th e natur e o f th e developmen t o f suc h analogie s i n many aspect s ha s been mos t difficult .

The first four chapter s o f thes e Lecture s dea l with th e theor y i n th e small . They are concerned wit h the analogue for functionals, o f the index of a critica l point o f th e functio n / . Conjugat e points, foca l points , characteristi c roots , the Poincar 6 rotation number , an d th e inde x o f concavit y o f close d extremal s are among the entities which serve to evaluate the index of a critical extremal, and which are unified by the theory of this index.

Chapter I V goe s beyond th e need s of th e theor y i n th e larg e in developin g separation, comparison, and oscillation theorems in w-spaee. Th e most general algebraic form o f linear , self-adjoin t boundar y condition s associate d wit h th e usual Jacob i differentia l equation s i s expose d i n a parametri c for m i n whic h only those coefficient s appear whic h ar e arbitrary . Th e theor y i s sufficientl y refined to specialize into a definite improvement upon the oscillation theorems of Bocher [2] and Ettlinger [1 , 2] in the 2-dimensional periodic case. Amon g other theorems, a necessar y an d sufficien t conditio n fo r th e existenc e o f infinitel y many characteristi c root s in our self-adjoin t boundar y problem s is established.

v

VI FOREWORD

Except for a theorem on the order of vanishing of the determinant of a conjugate family, most of the work of the first four chapters can be readily extended to the Bolza form of the Lagrange problem if the proper assumptions as to "normalcy" are made.

Chapter V presents the general boundary problem in the large. I t starts with a macroscopi c definitio n o f a Riemannia n manifol d R. Th e functiona l an d boundary conditions on R are defined in parametric form, and in the large. A first proble m which is solved concerns the invariantive or tensor definition of the indices of the preceding chapters. Thi s aspect of the theory will be of interest to differential geometers . Chapte r V treat s th e genera l accessor y boundar y problem in a way which is independent of the local coordinate systems employed. The author believes that this is the first general treatment of this character.

Chapter V I develop s th e theor y o f th e critica l point s o f a functio n o f m variables in a manner which seems best adapted to an extension to the case of functionals. Th e analogous treatment fo r the case of functiona l require s the development of the topology of the function space defined by a given boundary problem. Fo r problems for which the end points are always distinct, the func-tion space can be treated as in Chapter VII. Th e theory of the closed extremal in Chapter VIII requires a new approach to the topology of the corresponding function space . I n particular homologies which ar e not define d b y bounding are used here, and subgroups of substitutions of g points play an important rdle.

Chapter IX present s a solution of the Poinearg continuation problem which arose from Poinearf's study of Celestial Mechanics, Poincar6 [2]. Wit h Poincarg this problem reduced to the question of the existence and analytic continuation of a closed geodesic on a convex surface as the surface was varied analytically. Poincare* started with the principal ellipses on an ellipsoid. Th e validity of his reasoning has been questioned. I n Chapter IX explicit objections are presented. The presen t write r enlarge s th e Poincare * continuation proble m t o mea n th e problem of finding those numerical invariants of critical sets of closed extremals, the possession of whic h is a guarantee of the continued existence and analytic variation of critica l sets possessing the given numerical invariants as the basic Riemannian manifold is varied analytically. Thi s theory is applied to show that on an m-ellipsoid with unequal axes the principal ellipses vary analytically into critical sets of geodesies with the same numerical invariants, as the m-ellipsoid is varied analytically through a 1-parameter family of closed manifolds.

The author takes occasion here to acknowledg e his principal sources. Firs t of al l th e autho r wishe s t o acknowledg e hi s indebtednes s t o hi s colleague , Professor Georg e D . Birkhoff , whos e minima x principle , Birkhof f [1] , was the original stimulus of the present investigations, and whose transformation theory of dynamics, though logically less closely related to these Lectures, has by virtue of its broad aims and accomplishments proved no less inspiring. Th e author's knowledge of the classical theory has been acquired largely from the treatises of Bolza an d Hadamard , an d fro m th e work s o f Blis s whos e paper s o n th e n -dimensional theor y have been particularly useful . I n topology the author has

FOREWORD vn

been fortunat e i n havin g th e contemporar y wor k o f Veblen , Alexander , an d Lefschetz to follow, and to have had their papers always at his disposal.

The bibliography a t th e end o f th e Lectures is not intende d t o b e complete, but merel y t o lis t recen t paper s used b y th e author , o r paper s whic h ma y b e regarded as related to the work of the author.

The author acknowledges the generous aid furnished hi m by the Milton Fund of Harvard University for the preparation of the manuscript. Dr . S . B. Myers and Dr . A. W. Tucker have been kind enough to read part s of th e text an d t o offer valuabl e suggestions . Dr . Nanc y Col e ha s greatl y assiste d bot h i n th e reading and in the preparation of the manuscript.

The author extends his thanks to the American Mathematical Society and t o its officers for their invitation to present and publish these Lectures.

Cambridge, Massachusetts.

This page intentionally left blank

TABLE O F CONTENT S SECTION PAO B

FOREWORD — v

CHAPTER I

THE FIXE D EN D POIN T PROBLE M I N NON-PARAMETRI C FOR M

1. Th e Euler equations 1 2. Th e existence o f extremals 4 3. Th e necessary condition s of Weierstras s and Legendre 5 4. Th e Jacob i condition 7 5. Conjugat e point s 9 6. Th e Hilbert integra l 1 3 7. Sufficien t condition s 1 5

CHAPTER I I

GENERAL EN D CONDITION S

1. Th e end conditions 1 8 2. Th e transversality conditio n 2 0 3. Th e second variation 2 1 4. Th e accessory boundar y problem. 2 4 5. Th e necessary conditio n on the characteristi c root s 2 6 6. Th e non-tangency hypothesi s 2 8 7. Th e form Q(w , X) 3 0 8. Sufficien t condition s 3 3

CHAPTER II I

THE INDE X FOR M

1. Definitio n o f th e inde x form 3 7 2. Propertie s of th e inde x form. 4 2 3. Conjugat e families . 4 6 4. Necessar y conditions , on e end point variabl e 4 9 5. Foca l point s 5 1 6. Th e index of g in terms of foca l point s 5 5 7. Certai n lemmas on quadrati c form s 6 1 8. Tw o end manifolds 6 4 9. Periodi c extremals , a necessar y conditio n 7 0

10. Th e order of concavity — 7 1 11. Th e index o f a periodic extrema l — 7 4

CHAPTER I V

SELF-ADJOINT SYSTEM S

1. Self-adjoin t differentia l equation s 8 0 2. A representation o f self-adjoin t boundar y conditions 8 3 3. Boundar y problems involving a parameter 8 9 4. Compariso n o f problem s with differen t boundar y conditions 9 2 5. A general oscillatio n theore m 9 5

i x

x TABL E O F CONTENT S

6. Th e existenc e o f characteristi c roots . 9 7 7. Compariso n o f problems possessing different form s *> 9 9 8. Boundar y conditions at on e end alone 10 2

CHAPTER V

THE FUNCTIONA L O N A BIEMANNIA N SPAC E

1. A Riemannian space in the large 10 7 2. Basi c tensor s I l l 3. Th e necessary conditions o f Euler , Weierstrass , an d Legendre 11 3 4. Extremal s 11 5 5. Conjugat e point s 11 7 6. Th e Hilbert integra l - 11 9 7. Sufficienc y theorem s 12 0 8. Th e Jacobi equations i n tensor form 12 2 9. Th e genera l end conditions 12 6

10. Th e second variation 12 6 11. Th e accessory proble m in tensor form 12 7 12. Th e non-tangency conditio n 13 1 13. Characteristi c solution s in tensor form 13 3 14. Th e genera l inde x form. 13 7 15. Th e case o f en d manifolds 13 8

CHAPTER V I

THE CRITICA L BET S O F FUNCTION S

1. Th e non-degenerate case 14 2 2. Th e proble m of equivalence 14 6 3. Cycle s neighboring w 15 1 4. Neighborhoo d function s 15 2 5. Th e determination o f spannabl e and critical cycles . 15 6 6. Classificatio n o f cycles . 15 8 7. Th e type number s of a critica l se t 16 5 8. Justificatio n o f the coun t o f equivalen t critica l point s 17 5 9. Normal s from a point to a manifold 17 9

10. Symmetri c square s of manifolds 18 1 11, Critica l chord s of manifold s 18 3

CHAPTER VI I

THE BOUNDAR Y PROBLE M I N TH E LARG E

1. Th e functional domain 12 19 3 2. Th e functio n J(v) 19 6 3. Th e domai n J(w) < & . . . . . . 20 0 4. Restricte d domain s on fl 20 5 5. Th e /-distance betwee n restricte d curves 20 8 6. Cycle s on Q neighboring a critical se t m 21 2 7. Th e space 2 o f /-normal point s 21 3 8. Theore m 6.1 21 6 9. Cycle s on the domain s J < b and / < o 22 0

10. Th e existence of critica l extremal s 22 1 11. Th e non-degenerat e critica l extrema l 22 6 12. Th e non-degenerat e proble m - - • • 23 0 13. Th e fixed end point problem - • 23 4

TABLE O F CONTENT S X I

14. Th e one variable end point proble m 24 0 15. Th e two poin t functiona l connectivitie s o f a n m-sphere. 24 4

CHAPTER VII I

CLOSED EXTREMAL S

1. Th e complexes K, K*, an d IP 25 0 2. Th e infinite spac e Q. . . . . . - 25 3 3. Critica l sets o f extremal s — 25 6 4. Th e domain D* 25 8 5. Critica l set s on W 26 1 6. Critica l set s on a. 26 4 7. Th e extension o f a chain on n*. 27 3 8. Th e r-fold joi n o f a cycle 27 7 9. Finitenes s o f th e basi c maxima l set s 28 5

10. Numerica l invariant s o f a closed extrema l g 28 8 11. Th e non-degenerate close d extremal 29 1 12. Metric s with elementar y arc s — 29 7

CHAPTER I X

SOLUTION O F TH E POINCAK E CONTINUATIO N PROBLE M

1. Regula r submanifolds o f R p 30 7 2. Geodesie s o n m-ellipsoids 31 2 3. Th e indices of th e ellipses ga— — 31 6 4. Th e exclusiveness o f th e close d geodesies g\i 31 9 5. Th e linking cycles h\ t (a) 32 3 6. Symmetri c chain s and cycles' . — 32 6 7. Th e linking cycle s Xj,(o ) 33 3 8. Th e circular connectivities o f th e m-spher e , . 34 6 9. Topologicall y relate d closed extremal s 35 0

10. Continuatio n theorems 35 4 Bibliography 35 9 Index 36 7

This page intentionally left blank

BIBLIOGRAPHY ALEXANDER, J . W .

1. Combinatoria l analysi s situ s I . Transaction s o f th e America n Mathematica l Society 28 (1926), 301-329.

2. Combinatoria l theor y o f complexes . Annal s of Mathematic s 3 1 (1930), 294-322. ALEXANDROFF, P .

1. Einfachst e Grundbegriff e de r Topologie (mi t eine m Geleitwor t vo n Davi d Hilbert) . Berlin, Juliu s Springe r (1926) .

BlEBERBACH, L .

1. Theori e de r Differentialgleichungen . Berlin , Juliu s Springe r (1926) . BlBKHOFF, G . D .

1. Dynamica l system s wit h tw o degree s o f freedom . Transaction s o f th e America n Mathematical Society 18 (1917), 199-300.

2. Surfac e transformation s an d thei r dynamica l applications . Act a Mathematic a 4 3 (1920), 1-119.

3. Dynamica l systems . America n Mathematica l Societ y Colloquiu m Publication s 9 , New York (1927) .

4. O n the periodic motions of dynamical systems. Act a Mathematica 50 (1927), 359-379, 5. Stabilit y an d th e equation s o f dynamics . America n Journa l o f Mathematic s 4 9

(1927), 1-38 . 6. A new criterio n o f stability . Att i de l Congress o Internazional e de i Matematic i 6

(1928), 5-13. 7. Un e generalisatio n a n dimension s d u dernie r th^orem e d e g6om6tri e d e Poincarfc .

Comptes rendus des stances de PAead6mie des Sciences 192 (1931), 196-198. 8. (Wit h Lewis, D. C, Jr.) . O n the periodic motions near a given periodi c motion of a

dynamical system . Annal i d i Matematic a pur a e d applicat a 1 2 (1933-1934) , 117 -133.

9. Su r quelques courbes ferm6es remarquables. Bulleti n de la Soci6t6 Math&natique de France 60 , 1-26 .

BLISS, G . A .

1. (Wit h Mason , M.) . Th e propertie s o f curve s i n spac e whic h minimiz e a definit e integral. Transaction s o f th e America n Mathematica l Societ y 9 (1908) , 440-466.

2. (Wit h Mason , M.) . Field s o f extremal s i n space . Transaction s o f th e America n Mathematical Societ y 1 1 (1910), 325-340.

3. Th e Weierstras s ^-functio n fo r problem s o f th e calculu s o f variation s i n space . Transactions of the American Mathematica l Societ y 1 5 (1914), 369-378.

4. Jacobi' s conditio n fo r problem s o f th e calculu s o f variation s i n parametri c form . Transactions o f th e America n Mathematica l Societ y 1 7 (1916), 195-206 .

5. Th e proble m o f Maye r wit h variabl e en d points . Transaction s o f th e America n Mathematical Societ y 1 9 (1918), 305-314.

6. Th e transformatio n o f Clebsc h i n th e calculu s o f variations . Proceeding s o f th e International Mathematica l Congress , Toront o 1 (1924), 589-603.

7. Calculu s o f variations . Caru s Mathematica l Monograph . Chicag o (1925) . 8. A boundary value problem for a system of ordinary linear differential equation s of th e

first order . Transaction s o f th e America n Mathematica l Societ y 28 (1926) , 561-584.

9. A boundary valu e proble m i n th e calculu s o f variations . Bulleti n o f th e America n Mathematical Societ y 3 2 (1926), 317-331.

359

360 BIBLIOGRAPHY

10. Th e problem of Lagrange in the calculus of variations. America n Journal o f Mathe-matics 52 (1930), 674-743.

11. (Wit h Schoenberg , I . J.) . O n separation, compariso n an d oscillation theorem s for self-adjoint system s o f linea r secon d orde r differentia l equations . America n Journal o f Mathematics 53 (1931), 781-800.

12. Th e problem of Bolza in the calculus of variations. Annal s of Mathematics 33 (1932), 261-274.

13. (Wit h Hestenes, M. R.). Sufficien t condition s for a problem of Mayer in the calculus of variations . Transaction s o f th e America n Mathematica l Societ y 3 5 (1933) , 305-326.

B6CHER, M . 1. Introductio n to higher algebra. Ne w York (1912). 2. Lecon s sur les me^thodes de Sturm dan s la th^orie de s Equations diffe>entielle s line -

aires. Paris , Gauthier-Villars (1917) . BOLZA, O .

1. Vorlesunge n ube r Variationsrechnung . Leipzi g an d Berlin , B . G. Teubne r (1909) . 2. tlbe r de n "Abnormale n Fail " bei m Lagrangesche n un d Mayerschen Proble m mi t

gemischter Bedingunge n und variablen Endpunkten . Mathematisch e Annale n 74 (1913), 430-446.

BOYCE, M. G . 1. A n envelope theorem and necessary conditions for a problem of Mayer with variabl e

end points. Se e Chicago Theses (1930). BEOWN, A . B .

1. Relation s betwee n th e critical point s o f a real analyti c functio n o f N independen t variables. America n Journa l o f Mathematic s 5 2 (1930), 251-270.

2. Relation s betwee n th e critical point s an d curve s o f a rea l analyti c functio n o f two independent variables . Annal s o f Mathematic s 3 1 (1930) , 449-456.

3. Critica l sets of an arbitrary rea l analytic function o f n variables. Annal s of Mathe-matics 32 (1931), 512-S20.

4. Se e Koopman 1. CAIRNS, S . S .

1. Th e cellula r divisio n an d approximatio n o f regula r spreads . Proceeding s o f th e National Academ y of Sciences 16 (1930), 488-491.

2. O n th e cellula r subdivisio n o f n-dimensiona l regions . Annal s o f Mathematic s 3 3 (1932), 671-680.

CARATHEODORY, C.

1. tlbe r di e diskontinuierliche n Losunge n i n de r Variationsrechnung , Dissertation . Gottingen (1904) .

2. libe r geschlossene Extremalen un d periodische Variationsprobleme in der Ebene und im Raume. Annal i di Matematica (seri e IV) 2 (1924-25), 297-320.

3. Di e Methoden der geodatischen Xquidistante n und das problem von Lagrange. Act a Mathematics 47 (1926), 199-236.

4. Ube r die Existenz de r absoluten Minim a bei regulare n Variationsprobleme n au f der Kugel. Annal i della R. Scuola Normale Superior e di Pisa (serie II ) I (1932) , 79-87.

CHICAGO THESES 1. Contribution s t o th e calculu s o f variation s (1930 ) an d (1931-1932) . Universit y o f

Chicago Press. COPE, T . F .

1. Universit y of Chicago, Doctor's thesis (1927) . CoTOANT, R .

1. (Wit h Hilbert , D.) . Methode n de r Mathematisehe n Physik , I . Berlin , Juliu s Springer (1924) .

BIBLIOGRAPHY 361

CURRIER, A . E .

1. Th e variabl e en d poin t proble m o f th e calculu s o f variation s includin g a generaliza -tion o f th e classica l Jacob i conditions . Transaction s o f th e America n Mathemati -cal Societ y 3 4 (19S2) , 689-704 .

DAVIS , D . R .

1. Th e invers e proble m o f th e calculu s o f variation s i n highe r space . Transaction s o f the America n Mathematica l Societ y 3 0 (1928) , 710-736 .

DICKSON, L . E .

1. Moder n algebrai c theories . Ne w Yor k (1926) . DRESDEN, A .

1. Th e secon d derivative s o f th e extremal-integral . Transaction s o f th e America n Mathematical Societ y 9 (1908) , 467-486 .

EISENHART, L . P .

1. Riemannia n geometry . Princeto n (1926) . ETTLINGER, H . J .

1. Existenc e theorem s for the genera l rea l self-adjoint linea r system o f th e secon d order . Transactions o f th e America n Mathematica l Societ y 1 9 (1918) , 79-96 .

2. Oscillatio n theorem s fo r th e real , self-adjoin t linea r syste m o f th e secon d order . Transactions of th e America n Mathematica l Societ y 22 (1921) , 136-143 .

FRECHET, M .

1. Su r quelque s point s d u calcu l fonctionnel . Rendicont i de l Circol o Matematic o d i Palermo 2 2 (1906) , 1-74 .

GERGEN, J . J .

1. Mappin g o f a genera l typ e o f three-dimensiona l regio n o n a sphere . America n Journal o f Mathematic s 5 2 (1930) , 197-198 .

GRAVES, L . M .

1. Discontinuou s solution s i n th e calculu s o f variations . Bulleti n o f th e America n Mathematical Societ y 3 6 (1930) , 831-846 .

HADAMARD, J .

1. Lecon s sur le calcul de s variations , I . Paris , A . Herman n e t Fil s (1910) . H A H N , H .

1. t)be r raumlich e Variationsprobleme . Mathematisch e Annale n 7 0 (1911) , 110-142 . 2. t)be r ei n Existenztheore m de r Variationsrechnung . Sitzungsber . d . Akad . d . Wiss .

Wien, math.-naturw . Abt . Il- a 13 4 (1925) , 437-470 . HAUSDORFF, F .

1. Mengenlehre . Leipzig , Walte r d e Gruyte r (1927) . HEDLUND, G . A .

1. Poincar6' s rotatio n number and Morse's type number . Transaction s o f the America n Mathematical Societ y 3 4 (1932) , 75-97 .

HESTENES, M . R .

1. Se e Blis s 13. 2. Sufficien t condition s fo r th e genera l proble m o f Maye r wit h variabl e en d points .

Transactions o f th e America n Mathematica l Societ y 3 5 (1933) , 479-490 . HICKSON, A . O .

1. A n applicatio n o f th e calculu s o f variation s t o boundar y valu e problems . Transac -tions of th e America n Mathematica l Societ y 31 (1929) , 563-579 .

HILBERT, D .

1. Se e Couran t 1 . H O P F , H .

1. Vektorfelde r i n n-dimensionale n Mannigfaltigkeiten . Mathematisch e Annale n 9 6 (1926), 225-251.

362 BIBLIOGRAPHY

Hu, K U E N - S E N

1. Th e proble m o f Bolz a an d it s accessor y boundar y valu e problem . Se e Chicag o Theses (1931-1932) .

INCE, E . L .

1. Ordinar y differentia l equations . London , Longman s Gree n (1927) . JACKSON, D .

1. Algebrai c propertie s o f self-adjoin t systems . Transaction s o f th e America n Mathe -matical Societ y 1 7 (1916) , 418-424 .

JOHN, F .

1. t)be r di e Vollstandigkei t de r Relatione n vo n Mors e fu r di e Anzahle n kritische r Punkte. Mathematisch e Annale n 10 7 (1933), 1-14 .

JOHNSON, M . E .

1. Tensor s o f th e calculu s o f variations . America n Journa l o f Mathematic s 5 3 (1931) , 103-116.

KELLOGG, 0 . D .

1. Singula r manifold s amon g thos e o f a n analyti c family . Bulleti n o f th e America n Mathematical Societ y 3 5 (1929) , 711-716 .

v. KER£KJART6 , B .

1. Vorlesunge n fibe r Topologie , I . Flachentopologie . Berlin , Juliu s Springe r (1923) . KIANG, T S A I - H A N

1. O n the critical point s of non-degenerat e Newtonia n potentials . America n Journa l o f Mathematics 54 (1932), 92-109.

2. O n the existence of critical points of Green's function s fo r three-dimensiona l regions . American Journa l o f Mathematic s 5 4 (1932) , 657-666 .

3. Critica l points of harmonic functions an d Green's functions i n plane regions. Scienc e Quarterly of the National University of Peking 3 , 113-123 .

KNESER, A .

1. Lehrbuchde r Variationsrechnung , II . Braunschwei g (1925) . KOOPMAN, B . O .

1. (Wit h Brown , A . B.) . O n the coverin g o f analyti c loc i b y complexes . Transaction s of th e America n Mathematica l Societ y 3 4 (1932) , 231-251 .

KRONECKER, L .

1. Ube r System e vo n Funktione n mehre r Variabeln . Werke , I , 175-226 . 2. tlbe r di e Charakteristi k vo n Funktionen-Systemen . Werke , II , 71-82 .

LAREW, G . A .

1. Th e Hilber t integra l an d Maye r fields fo r th e proble m o f Maye r i n th e calculu s o f variations. Transaction s o f th e America n Mathematica l Societ y 2 6 (1924) , 61-67 .

LEFSCHETZ, S .

1. Topology . America n Mathematica l Societ y Colloquiu m Publication s 12 . Ne w York (1930) .

2. O n singula r chain s an d cycles . Bulleti n o f th e America n Mathematica l Societ y 3 9 (1933), 124-129 .

3. Continuou s transformation s o f manifolds . Proceeding s o f th e Nationa l Academ y o f Sciences 1 1 (1925) , 290-292 .

LEWIS, D . C , JR .

1. Se e Birkhof f 8 . LINDENBATJM, A .

1. Contribution s a 1'etude d e l'espace m^trique , I . Fundament a Mathematica e 8 (1926) , 209-222.

LITTAUER, S . B .

1. Se e Mors e 18 . LUSTERNIK, L .

1. (Wit h Schnirrelmann, L.) . Topologica l method s in variational problems . Researc h Institute o f Mathematic s an d Mechanics . Gozida t Mosco w (1930) .

BIBLIOGRAPHY 363

MCSHANE, E . J .

1. Semi-continuit y i n the calculus of variations, and absolute minima fo r isoperimetri c problems. Contribution s t o th e calculu s o f variations . Universit y o f Chicag o (1930).

MASON, M .

1. Se e Bliss 1,2 . MAYER, W .

1. Riemannsch e Geometri e (Lehrbuc h de r Differentialgeometrie , II , vo n Duschek , A . und Mayer, W.). Leipzi g (1930).

MENGER, K .

1. Uebe r geodatische Linien in allgemeinen metrischen Raumen. Koninklijk e Akademie van Wetenschappe n t e Amsterdam 2 9 (1926), 1-4 .

2. Untersuchunge n libe r allgemein e Metrik . Viert e Untersuchung . Zu r Metri k der Kurven . Mathematisch e Annale n 10 3 (1930), 466-501.

3. Berich t iibe r metrisch e Geometrie . Jahresberich t de r Deutsche n Mathematiker -Vereinigung 40 (1931), 201-219.

4. Som e applications o f poin t se t methods . Annal s of Mathematics 3 2 (1931), 739-760. MORSE, M .

1. Relation s betwee n th e critica l point s o f a rea l functio n o f n independen t variables . Transactions of the American Mathematical Societ y 27 (1925), 345-396.

2. Th e analysis and analysis situs of regular n-spreads in (n -f r)-space . Proceeding s of the National Academy of Sciences 13 (1927), 813-817.

3. Th e foundations o f a theory of the calculus of variation s in the large. Transaction s of the American Mathematical Societ y 30 (1928), 213-274.

4. Singula r point s o f vecto r fields unde r genera l boundar y conditions . America n Journal o f Mathematics 51 (1929), 165-178.

5. Th e foundation s o f th e calculu s o f variation s i n th e larg e i n m-spac e (firs t paper) . Transactions o f th e America n Mathematica l Societ y 31 (1929) , 379-404.

6. Th e critical point s of functions an d the calculus of variations in the large . Bulleti n of the American Mathematical Societ y 35 (1929), 38-54.

7. Close d extremals. Proceeding s of the Nationa l Academ y o f Science s 1 5 (1929), 856-859.

8. Th e problems of Lagrange and Mayer under genera l end conditions . Proceeding s of the National Academy of Sciences 16 (1930), 229-233.

9. Th e foundation s o f a theor y o f th e calculu s o f variation s i n th e larg e i n m-spac e (second paper) . Transaction s o f th e America n Mathematica l Societ y 3 2 (1930) , 599-631.

10. A generalizatio n o f th e Stur m separatio n an d compariso n theorem s i n n-space . Mathematische Annalen 10 3 (1930), 52-69.

11. Th e critica l point s o f a function o f n variables . Proceeding s o f the Nationa l Acad -emy of Sciences 16 (1930), 777-779.

12. Th e critica l point s o f a functio n o f n variables . Transaction s o f th e America n Mathematical Societ y 33 (1931), 72-91.

13. Th e orde r o f vanishin g o f th e determinan t o f a conjugate base . Proceeding s o f th e National Academy of Sciences 17 (1931), 319-320.

14. Sufficien t condition s i n th e proble m o f Lagrang e wit h fixe d en d points . Annal s o f Mathematics 32 (1931), 567-577.

15. (Wit h Myers, S. B.). Th e problems of Lagrange and Mayer with variable end points. Proceedings of the American Academy of Art s and Sciences 66 (1931), 235-253.

16. Sufficien t condition s in the problem of Lagrange with variable end conditions. Amer -ican Journal of Mathematics 53 (1931), 517-546.

17. Close d extremal s (firs t paper) . Annal s o f Mathematic s 3 2 (1931) , 549-566 .

364 BIBLIOGRAPHY

18. (Wit h Littauer , S . B.) . A characterizatio n o f fields i n th e calculu s o f variations . Proceedings of the National Academy of Sciences 18 (1932), 724-730.

10. Th e calculus of variations in the large. Verhandlunge n de s In t e rna t iona l Mathe -matiker-Kongresses, Zuric h (1032) .

20. (Wit h va n Schaack , G . B.) . Th e critica l poin t theor y unde r genera l boundar y con -ditions. Annal s of Mathematic s 35 (1034).

21. (Wit h Pitcher , E,) . O n certain invariant s of close d extremals . Proceeding s o f th e National Academ y of Sciences 20 (1034), 282-288.

MURNAGHAN, F. D . 1. Vecto r analysis and the theory of relativity. Baltimor e (1022) .

MYERS, S . B . 1. Se e Morse 15. 2. Adjoin t system s in the problem of Mayer under general end conditions . Bulleti n o f

the America n Mathematica l Societ y 38 (1032), 303-312. 3. Sufficien t condition s in th e problem of th e calculus of variation s in n-spaee i n para -

metric for m an d unde r genera l en d conditions . Transaction s o f th e America n Mathematical Societ y 35 (1033), 746-760.

OSGOOD, W . F . 1. Lehrbuc h de r Funktionentheorie , II . Leipzig , B . G. Teubne r (1020) .

PITCHER, E . 1. Se e Morse 21.

PLANCHEREL, M . 1. Su r l a ni6thod e d 1 integration d e Ritz . Bulleti n de s Science s Mathematique s 4 8

(1024); 12-48, 58-80, 03-100. POINCARE, H .

1. Su r les courbes d^finie s pa r le s equations differentieiles , par t 3 . Liouviil e Journa l (4th series ) 1 (1885), 167-244 .

2. Su r le s ligne s g6od6sique s de s surface s convexes . Transaction s o f th e America n Mathematical Societ y 6 (1005) , 237-274 .

3. Mdthode s nouvelles de la mScanique c61este, 1-3. Paris , Gauthier-Villars (1802) . PRICE, G . B .

1. A class of dynamical systems on surfaces of revolution. America n Journal o f Mathe-matics 5 4 (1032) , 753-768 .

RADON, J . 1. tJbe r da s Minimu m de s Integrals . Sitzungsbericht e d . Akad . d . Wiss . Wien, math. -

naturw. Abt . Il- a 11 0 (1010), 1257-1326. 2. tlbe r die Oszillationstheoreme de r konjugierten Punkt e beim Problem von Lagrange .

Sitzungsberichte de r math.-naturw . Abteilun g de r Bayerische n Akademi e de r Wissenschaften z u Miinchen (1027) , 243-257.

REID, W . T . 1. A boundary valu e proble m associate d wit h th e calculu s o f variations . America n

Journal o f Mathematics 5 4 (1032), 760-780. 2. O n boundar y valu e problem s associate d wit h doubl e integral s i n th e calculu s o f

variations. America n Journal o f Mathematics 54 (1032), 701-801. RICHARDSON, R . G . D .

1. Da s Jacobische Kriterium de r Variationsrechnung un d die Oszillationseigenschafte n linearer Differentialgieichunge n 2 . Ordnung. Mathematisch e Annale n 6 8 (1010) , 270-304.

RICHMOND, D . E . 1. Numbe r relation s betwee n type s o f extremal s joinin g a pai r o f points . America n

Journal of Mathematics 50 (1028), 371-388. 2. A new proo f o f certai n relation s o f Mors e i n th e calculu s o f variation s i n th e large .

Bulletin o f the American Mathematica l Societ y 35 (1020), 248-258.

BIBLIOGRAPHY 365

ROZENBERG, J .

1. Ube r das Verhaiten von Extremalenbogen di e de n zu m Anfangspunk t konjugierte n Punkt enthalte n bei m Lagrangesche n Proble m de r Variationsrechnung . Monat -shefte fu r Mathemati k un d Physik 2 4 (1913), 65-86.

VAN SCHAACK, G . B . 1. Se e Morse 20.

SCHNIRRELMANN, L . 1. SeeLusterni k 1 .

SCHOENBERG, I. J. 1. Se e Bliss 11.

SlGNORlNI, A .

1. Esistenz a di un'estremale chius a dentro u n eontorno di Whittaker . Rendicont i de l Circolo Matematico d i Palermo 33 (1912), 187-193.

SMITH, P . A .

1. Th e topolog y o f involutions . Proceeding s o f th e Nationa l Academ y o f Science s 1 9 (1933), 612-618.

STRUIK, D . J .

1. Differentia l geometr y in th e large. Bulleti n o f the America n Mathematica l Societ y 37 (1931) , 49-62 .

STURM, C .

1. MSmoir e sur les equations differentielles lineaire s du second ordre. Liouvill e Journa l 1 (1836), 106-186.

TONELLI, L . 1. Fondament i d i calcolo dell e variazioni , I an d II . Bologna , Nicol a Zanichelli (1921 )

and (1923) . TUCKER, A . W .

1. A n abstract approac h to manifolds. Annal s of Mathematics 34 (1933), 191-243. 2. O n tensor invariance in the calculus of variations. Annal s of Mathematics 35 (1934),

VEBLEN, O .

1. Analysi s situs. America n Mathematica l Societ y Colloquium Publication s 5 , par t 2 . New York (1931) .

2. (Wit h Whitehead, J . H. C ). A set of axioms for differential geometry . Proceeding s of the National Academy of Sciences 17 (1931), 551-561.

3. (Wit h Whitehead , J . H . C ) . Th e foundation s o f differentia l geometry . Tendon , Cambridge University Pres s (1932).

VOLTERRA, V .

1. Theor y o f functiona l an d integra l an d integro-differentia l equations . Translate d by Miss M. Long. London , Blackie and Son (1930).

VAN DER WAERDEN, B . L . 1. Topologisch e Begriindun g des Kalkiils de r abzahlende n Geometri c Mathcmatisch c

Annalen 102 (1929), 337-362. WALSH, J . L .

1. Not e o n th e locatio n o f th e critica l point s o f Green' s function . Bulleti n o f th e American Mathematical Societ y 39 (1933), 775-783.

WHITE, M . B .

I. Th e dependence of focal point s upon curvature for problems o f th e calculus of varia -tions i n space . Transaction s o f th e America n Mathematica l Societ y 1 3 (1912) , 175-198.

WHITEHEAD, J . H . C .

1. Se e Veble n 2 , 3. 2. Th e Weierstras s ^-functio n i n differentia l geometry . Th e Quarterl y Journa l o f

Mathematics (Oxfor d Series ) 4 (1933), 291-296.

366 BIBLIOGRAPHY

WHYBURN, W. M . 1, Non-isolate d critica l point s o f functions . Bulleti n of th e American Mathematical

Society 35 (1029), 701-708. WlNTNBR, A .

1. Ube r di e Jacobisch e Differentialgleichun g de s restringierte n Dreikorperprobleme . Sitzungsberichte de r math.-phys. Klass e de r S&chsische n Akademi e de r Wissen -echaften zu Leipzig 82 (1930), 345-354.

2. Thre e note s o n characteristi c exponent s an d equation s o f variatio n in celestia l mechanics. America n Journal o f Mathematics 5 3 (1931), 605-625.

INDEX

Numbers refer t o t

Abel, 313. Alexander, vii , 146 , 167 , 182 , 359. Alexandroff, 146 , 359.

Behaghel, 107 , 122. Bieberbach, 359 . Birkhoff, vi , 142 , 143, 192, 305, 306, 307, 359. Bliss, vi , 1 , 3 , 7 , 8 , 18 , 19 , 36 , 64 , 80 , 107 ,

113, 359. Bdeher, v , 83 , 96, 97 , 110 , 113 , 360. Bolza, vi , 1 , 19 , 36, 46, 64, 80, 113 , 192, 313,

360. Boyce, 360 . Brown, 167 , 198 , 360.

Cairns, 360 . CarathSodory, 4 , 18 , 79, 192 , 360. Chicago Theses, 18 , 360. Cole, vii . Cope, 36 , 360. Courant, 36 , 62, 80, 145 , 360. Currier, 18 , 64, 361.

Davis, 81 , 361. Dickson, 32 , 361. Dresden, 361 . Du Bois-Reymond, 2 .

Eisenhart, 111 , 152 , 361. Erdmann, 3 . von Escherich , 46 , 52 , 103 . Ettlinger, v , 96 , 361. Euler, 1 , 19 , 113.

Freehet, 209 , 297, 354 , 361.

Gergen, 361. Graves, 4 , 361.

Hadamard, vi , 1 , 79, 113 , 361. Hahn, 61 , 361. Hausdorff, 361 . Hedlund, 79 , 307 , 361. Hestenes, 361 . Hickson, 80 , 361.

appropriate page s

Hilbert, v , 3 , 13 , 36 , 62 , 80 , 119 , 120 , 192 , 361.

Hopf, 144 , 361. Hu, Kuen-Sen , 80 , 362.

Ince, 95 , 96 , 362.

Jackson, 362 . Jacobi, 7 , 8 , 10 , 16 , 107 , 120 , 122 , 125 , 313. John, 145 , 362. Johnson, 362 . Jordan, 172 .

Kellogg, 178 , 362. v. Kereltjarta , 362 . Kiang, Tsai-Han , 142 , 362. Kneser, 362 . Koopman, 198 , 362. Kronecker, 9 , 32 , 145 , 362.

Lagrange, vi , 36 , 52, 80, 110 , 173. Larew, 362. Lefschetz, vii , 107 , 144 , 146 , 182 , 252, 362. Legendre, 5 , 6 , 16 , 113 , 114 , 118 , 120 . Lewis, 362 . Lindenbaum, 297 , 362. Liouville, 20 , 102. Littauer, 362. Lusternik, 305 , 307, 362.

McShane, 192 , 363. Mason, 3 , 363. Mayer, A. , 11 , 13 , 120. Mayer, W. , 363. Menger, 299 , 363. Morse, 18 , 28 , 36 , 37 , 45 , 47 , 61 , 62, 64 , 75 ,

78, 80 , 99 , 104 , 110 , 143 , 145 , 163 , 180 , 191, 305, 363.

Murnaghan, 364 . Myers, vii , 18 , 28, 364.

Osgood, 19 , 198 , 257, 364 .

Pitcher, 364 . Plancherel, 80 , 364.

368 INDEX

Poincare, v , vi , 19 , 79 , 143 , 305 , 306 , 354 , 364.

Price, 364 .

Radon, 364 . Rcid, 364 . Richardson, M. , 191. Richardson, R . D . G. , 80 , 364. Richmond, 192 , 364. Rosenberg, 61 , 365.

van Schaaek , 143 , 145, 365. Schnirrelmann, 305 , 307, 365. Schoenberg, 365. Signorim, 192 , 365. Smith, 191 , 365.

Struik, 365 . Sturm, 20 , 78, 80, 102 , 365.

Tonelli, 192 , 365. Tucker, vii , 131 , 146, 365.

Veblen, vii , 107 , 110 , 167 , 182 , 365. Volterra, 365 .

van de r Waerden , 365. Walsh, 365. Weierstrass, 3 , 5 , 15 , 16 , 112 , 113 , 114 , 120 . White, 365. Whitehead, 110 , 365. Whyburn, 145 , 366. Wintner, 366 .

Titles i n Thi s Serie s (Continued from the front of this publication)

W . F . Osgood , Topic s i n th e theor y o f function s o f severa l comple x variables , 191 4

3.1 G . A . Bl iss , Fundamenta l existenc e theorems , 191 3

3.2 E . Kasner , Differential-geometri c aspect s o f dynamics , 191 3

2 E . H . M o o r e , Introductio n t o a for m o f genera l analysi s

M. Mason , Selecte d topic s i n th e theor y o f boundar y valu e problem s o f differentia l equations

E. J . Wilczyriski , Projectiv e differentia l geometry , 191 0

1 H . S . W h i t e , Linea r system s o f curve s o n algebrai c surface s

F. S . W o o d s , Form s o n noneuclidea n spac e

E. B . Va n Vleck , Selecte d topic s i n th e theor y o f divergen t serie s an d o f continue d fractions, 190 5